On a Vizing-like conjecture for direct product graphs

Department of Mathematics, PEF, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
Discrete Mathematics (Impact Factor: 0.57). 09/1996; 156(1-3):243-246. DOI: 10.1016/0012-365X(96)00032-5
Source: DBLP

ABSTRACT Let fl(G) be the domination number of a graph G, and let G \Theta H be thedirect product of graphs G and H . It is shown that for any k 0 there existsa graph G such that fl(G \Theta G) fl(G)

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    ABSTRACT: The dual notions of domination and packing in finite simple graphs were first exten-sively explored by Meir and Moon in [15]. Most of the lower bounds for the domination number of a nontrivial Cartesian product involve the 2-packing, or closed neighborhood packing, number of the factors. In addition, the domination number of any graph is at least as large as its 2-packing number, and the invariants have the same value for any tree. In this paper we survey what is known about the domination, total domina-tion and paired-domination numbers of Cartesian products and direct products. In the process we highlight two other packing invariants that each play a role similar to that played by the 2-packing number in dominating Cartesian products.
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    ABSTRACT: A sharp lower bound for the domination number and the total domination number of the direct product of finitely many complete graphs is given: γ(×i=1tKni)≥t+1,t≥3. Sharpness is established in the case when the factors are large enough in comparison to the number of factors. The main result gives a lower bound for the domination (and the total domination) number of the direct product of two arbitrary graphs: γ(G×H)≥γ(G)+γ(H)−1γ(G×H)≥γ(G)+γ(H)−1. Infinite families of graphs that attain the bound are presented. For these graphs it also holds that γt(G×H)=γ(G)+γ(H)−1γt(G×H)=γ(G)+γ(H)−1. Some additional parallels with the total domination number are made.
    Discrete Mathematics 12/2010; 310(23):3310–3317. DOI:10.1016/j.disc.2010.07.015 · 0.57 Impact Factor


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